1. Discrete Mathematics Relations

2. What is a relation Relation generalizes the notion of functions.
Recall: A function takes EACH element from a set and maps it to a UNIQUE element in another set
f: X  Y
 x  X,  y such that f(x) = y
Let A and B be sets.
A binary relation R from A to B is a subset of A  B
Recall: A x B = {(a, b) | a  A, b  B}
aRb: (a, b)  R.
Application
Relational database model is based on the concept of relation.

3. What is a relation Example Let A be the students in a the CS major
A = {Ayşe, Barış, Canan, Davut}
Let B be the courses the department offers
B = {BİM111, BİM122, BİM124}
We specify relation R  A  B as the set that lists all students a  A enrolled in class b  B
R = {(Ayşe, BİM111), (Barış, BİM122), (Barış, BİM124), (Davut, BİM122), (Davut, BİM124)}

4. More relation examples
Another relation example:
Let A be the cities in Turkey
Let B be the districts in Turkey
We define R to mean a is a district in city b
Thus, the following are in our relation:
(Bakırköy, İstanbul)
(Keçiören, Ankara)
(Nilüfer, Bursa)
(Tepebaşı, Eskişehir)
etc…
Most relations we will see deal with ordered pairs of integers

5. Representing relations
We can represent relations graphically:
We can represent relations in a table:
BİM111
BİM122
BİM124
Ayşe X
Barış
Canan
Davut
BİM111
BİM122
BİM124
Ayşe
Barış
Canan
Davut
Not valid functions!

6. Relations vs. functions
If R  X  Y is a relation, then is R a function?
If f: X  Y is a function, then is f a relation?

7. Relations on a set A relation on the set A is a relation from A to A
In other words, the domain and co-domain are the same set
We will generally be studying relations of this type

8. Relations on a set R 1 2 3 4 X Let A be the set { 1, 2, 3, 4 }
Which ordered pairs are in the relation
R = { (a,b) | a divides b }
R = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)} 1 2 3 4 1 2 3 4 R 1 2 3 4 X

9. More examples Consider some relations on the set Z
Are the following ordered pairs in the relation?
(1,1) (1,2) (2,1) (1,-1) (2,2)
R1 = { (a,b) | a≤b }
R2 = { (a,b) | a>b }
R3 = { (a,b) | a=|b| }
R4 = { (a,b) | a=b }
R5 = { (a,b) | a=b+1 }
R6 = { (a,b) | a+b≤3 } X X X X X X X X X X X X X X X

10. Relation properties Six properties of relations we will study:
Reflexive
Irreflexive
Symmetric
Asymmetric
Antisymmetric
Transitive

11. Reflexivity vs. Irreflexivity
Definition: A relation is reflexive if
(a,a)  R for all a  A
Irreflexivity
Definition: A relation is irreflexive if
(a,a)  R for all a  A
Examples
Is the “divides” relation on Z+ reflexive?
Is the “” (not ) relation on a P(A) irreflexive? =
<
>  
reflexive o x x o o
irreflexive x o o x x

12. Reflexivity vs. Irreflexivity
A relation can be neither reflexive nor irreflexive
Example?
A = {1, 2}, R = {(1, 1)}
It is not reflexive, since (2, 2)  R,
It is not irreflexive, since (1, 1)  R.

13. Symmetry, Asymmetry, Antisymmetry
A relation is symmetric if
for all a, b  A, (a,b)  R (b,a)  R
A relation is asymmetric if
for all a, b  A, (a,b)  R  (b,a)  R
A relation is antisymmetric if
for all a, b  A, ((a,b)  R  (b,a)  R)  a=b
(Second definition) for all a, b  A, ((a,b)  R  a  b)  (b,a)  R)
<
> =  
isTwinOf
symmetric x x o x x o
asymmetric o o x x x x
antisymmetric o o o o o x

14. Notes on *symmetric relations
A relation can be neither symmetric or asymmetric
R = { (a,b) | a=|b| }
This is not symmetric
-4 is not related to itself
This is not asymmetric
4 is related to itself
Note that it is antisymmetric

15. Transitivity A relation is transitive if
for all a, b, c  A, ((a,b)R  (b,c)R)  (a,c)R
If a < b and b < c, then a < c
Thus, < is transitive
If a = b and b = c, then a = c
Thus, = is transitive

16. Transitivity examples
Consider isAncestorOf()
Let Ayşe be Barış’s ancestor, and Barış be Canan’s ancestor
Thus, Ayşe is an ancestor of Barış, and Barış is an ancestor of Canan
Thus, Ayşe is an ancestor of Canan
Thus, isAncestorOf() is a transitive relation
Consider isParentOf()
Let Ayşe be Barış’s parent, and Barış be Canan’s parent
Thus, Ayşe is a parent of Barış, and Barış is a parent of Canan
However, Ayşe is not a parent of Canan
Thus, isParentOf() is not a transitive relation

17. Summary of properties of relations
reflexive
a (a, a)  R
irreflexive
a (a, a)  R
symmetric
 a, b  A, (a,b)  R  (b,a)  R
asymmetric
 a, b  A, (a,b)  R  (b,a)  R
antisymmetric
 a, b  A, ((a,b)  R  (b,a)  R)  a=b
(*)for all a, b  A, ((a,b)  R  a  b)  (b,a)  R)
transitive
 a, b, c  A, ((a,b)  R  (b,c)  R)  (a,c)  R
(*) Alternative definition…

18. Combining relations There are two ways to combine relations R1 and R2
Via Set operators
Via relation “composition”

19. Combining relations via Set operators
Consider two relations R≥ and R≤
R≥ U R≤ = all numbers ≥ OR ≤
That’s all the numbers
R≥ ∩ R≤ = all numbers ≥ AND ≤
That’s all numbers equal to
R≥  R≤ = all numbers ≥ or ≤, but not both
That’s all numbers not equal to
R≥ - R≤ = all numbers ≥ that are not also ≤
That’s all numbers strictly greater than
R≤ - R≥ = all numbers ≤ that are not also ≥
That’s all numbers strictly less than
Note that it’s possible the result is the empty set

20. Combining via relational composition
Similar to function composition
Let R be a relation from A to B, and S be a relation from B to C
Let a  A, b  B, and c  C
Let (a,b)  R, and (b,c)  S
Then the composite of R and S consists of the ordered pairs (a,c)
We denote the relation by S ◦ R
Note that S comes first when writing the composition!
(a, c)  S ◦ R if  b such that (a, b)  R, and (b,c)  S

21. Combining via relational composition
Let M be the relation “is mother of”
Let F be the relation “is father of”
What is M ◦ F?
If (a,b)  F, then a is the father of b
If (b,c)  M, then b is the mother of c
Thus, M ◦ F denotes the relation “maternal grandfather”
What is F ◦ M?
If (a,b)  M, then a is the mother of b
If (b,c)  F, then b is the father of c
Thus, F ◦ M denotes the relation “paternal grandmother”
What is M ◦ M?
Thus, M ◦ M denotes the relation “maternal grandmother”

22. Combining via relational composition
Given relation R
R ◦ R can be denoted by R2
R2 ◦ R = (R ◦ R) ◦ R = R3
Example: M3 is your mother’s mother’s mother